Problem: $h(x)=\dfrac{10}{x-6}$. On which intervals is the graph of $h$ concave up? Choose 1 answer: Choose 1 answer: (Choice A) A $x<6$ only (Choice B) B $x>\dfrac53$ only (Choice C) C $x<\dfrac53$ only (Choice D) D $x>6$ only
Solution: We can analyze the intervals where $h$ is concave up/down by looking for the intervals where its second derivative $h''$ is positive/negative. This analysis is very similar to finding increasing/decreasing intervals, only instead of analyzing $h'$, we are analyzing $h''$. The second derivative of $h$ is $h''(x)=\dfrac{20}{(x-6)^3}$. $h''$ is never equal to $0$. $h''$ is undefined for $x=6$. Therefore, our only point of interest is $x=6$. Our point of interest divides the domain of $h$ (which is all numbers except for $6$ ) into two intervals: $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $x<6$ $x>6$ Let's evaluate $h''$ at each interval to see if it's positive or negative on that interval. Interval $x$ -value $h''(x)$ Verdict $x<6$ $x=5$ $h''(5)=-20<0$ $h$ is concave down $\cap$ $x>6$ $x=7$ $h''(7)=20>0$ $h$ is concave up $\cup$ In conclusion, the graph of $h$ is concave up over the interval $x>6$ only.